If t→ 0(∫₀¹ (3x + 5)^t dx)(1)/(t) = (α)/(5e)((8)/(5))(2)/(3)$\lim_{t\to 0}\left(\int_0^1 (3x + 5)^t dx\right)^{\frac{1}{t}} = \frac{\alpha}{5e}\left(\frac{8}{5}\right)^{\frac{2}{3}}$, then α$\alpha$ is equal to

Related Formula

For a limit of 1∞$1^{\infty}$ form, where f(x) = 1$\lim f(x) = 1$ and g(x) = ∞$\lim g(x) = \infty$:

Taylor expansion of x$\tan x$:

Core Logic

Here, x → 0 ( x)/(x) = 1$\lim_{x \to 0} \frac{\tan x}{x} = 1$ and x → 0 (1)/(x²) = ∞$\lim_{x \to 0} \frac{1}{x^2} = \infty$.

Step 1: Finding limit exponent

Using expansion of x$\tan x$:

Thus: p = e1/3$p = e^{1/3}$

Pattern Recognition

Limits of forms like 1∞$1^{\infty}$ always boil down to evaluating standard polynomials in the exponent. Using Taylor series expansion rather than L'Hopital's rule directly avoids taking multiple heavy derivatives.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability

Related Formula

Standard limits commands:

Core Logic

We can rewrite the limit by grouping each term with its standard balancing factor:

Step 1: Simplify the Compensating Factor

Evaluate the remaining algebraic factor:

Since all individual standard limit terms approach 1$1$, the value of the limit is exactly:

Pattern Recognition

Shortcut: For standard limits involving (u)$\tan(u)$, ln(1+u)$\ln(1+u)$, ⁻¹(u)$\tan^{-1}(u)$, and e^u-1$e^u-1$ as u → 0$u \to 0$, replace each function directly with its argument:

Chapter Mix

Class 11 Mathematics: Limits and Derivatives Class 12 Mathematics: Limits, Continuity and Differentiability

Related Formula

The greatest integer function [u]$[u]$ changes value and experiences a step discontinuity at any point where its inner argument u$u$ takes on an integer value.

Core Logic

Analyze the potential points where either component function argument changes into an integer within the interval x in [0, 4]$x \in [0, 4]$.

• For [(x²)/(2)]$\left[\frac{x^2}{2}\right]$:

(x²)/(2)$\frac{x^2}{2}$ can range from (0)/(2) = 0$\frac{0}{2} = 0$ up to (16)/(2) = 8$\frac{16}{2} = 8$. Integer values are reached at (x²)/(2) = 0, 1, 2, 3, 4, 5, 6, 7, 8$\frac{x^2}{2} = 0, 1, 2, 3, 4, 5, 6, 7, 8$, which means critical test locations are:

• For [√(x)]$[\sqrt{x}]$:

√(x)$\sqrt{x}$ can range from √(0) = 0$\sqrt{0} = 0$ to √(4) = 2$\sqrt{4} = 2$. Integer values are reached at √(x) = 0, 1, 2$\sqrt{x} = 0, 1, 2$, which means critical test locations are:

Step 1: Audit Each Critical Point

Combine the set of test points within domain boundaries (0, 4)$(0, 4)$:

Let's evaluate the left and right hand limits at these specific values:

• At x = 1$x = 1$: [√(x)]$[\sqrt{x}]$ steps up while [(x²)/(2)]$\left[\frac{x^2}{2}\right]$ is constant $\implies$ Discontinuous.
• At x = √(2)$x = \sqrt{2}$: [(x²)/(2)]$\left[\frac{x^2}{2}\right]$ steps up while [√(x)]$[\sqrt{x}]$ is constant $\implies$ Discontinuous.
• At x = 2$x = 2$: Both functions experience an simultaneous integer step. Let's inspect:
• f(2) = [2] - [√(2)] = 2 - 1 = 1$f(2) = [2] - [\sqrt{2}] = 2 - 1 = 1$
• f(2^-) = [1.99] - [1.41] = 1 - 1 = 0$f(2^-) = [1.99] - [1.41] = 1 - 1 = 0$

Since LHL ≠$\neq$ value at point, it is Discontinuous. Continuing this verification down the full combined list confirms that none of the step jumps cancel each other out.

Step 2: Sum the Discontinuity Points

Counting all isolated inner points within (0, 4)$(0, 4)$ yields exactly 8$8$ locations:

Pattern Recognition

When two greatest integer functions drop steps simultaneously at the same point (like at x=2$x=2$), always write out the explicit left and right limits manually, as simultaneous steps occasionally step in matching directions and maintain unexpected continuity.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability

Related Formula

Core Logic

Verify Statement I via high-order Taylor polynomial series tracking, and determine Statement II values by resolving standard exponential limit properties.

Step 1: Expand Statement I Sequence Polynomials

Summing terms together and subtracting 2x$2x$ leaves:

Step 2: Verify Statement II Limit Power Structure

Evaluating the 1^∞$1^\infty$ form configuration style:

Step 3: State Conclusion

Both statements are correct.

Pattern Recognition

When a limit features a power factor of 5 in the denominator, you must track expansion variables through the 5th degree term to guarantee accuracy.

Chapter Mix

Class 11 Mathematics: Limits

Core Logic

The expression |x|$\cos|x|$ is everywhere differentiable. Thus, non-differentiability relies completely on the modulus function containing the quadratic factor, i.e., |x² - ax + 2|$|x^2 - ax + 2|$.

Non-differentiability points generally happen where:

Step 1: Evaluation of Roots

Given one of the roots is α = 2$\alpha = 2$:

Substituting a=3$a=3$ gives the other root β = 1$\beta = 1$. However, evaluating differentiability at x=1$x=1$ reveals properties that invalidate standard options.

Step 2: Conclusion

Due to a technical contradiction in the configuration of the differentiable constraints at x=1$x=1$, this question was officially dropped by NTA.

Pattern Recognition

If a quadratic inside a modulus has distinct real roots, it normally creates non-differentiable sharp turns unless a repeated multiplying factor outside cancels it out.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability